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Basic Algebra and Graphing for Electric Circuits
16 questions By Tony R. Kuphaldt
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Question 4 of 16
Observe the following equivalence:
43 42= 4 ×4 ×4 4 ×4It should be readily apparent that we may cancel out two quantities from both top and bottom of the fraction, so in the end we are left with this:
4 1Re-writing this using exponents, we get 41.
Expand each of these expressions so that there are no exponents either:
\((\frac{3^5}{3^2})\) =
\((\frac{10^6}{10^4})\) =
\((\frac{8^7}{8^3})\) =
\((\frac{20^5}{20^4})\) =
After expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (41). From these examples, what pattern do you see with exponents of products. In other words, what is the general solution to the following expression?
am an= Reveal answeram an= am−n Notes:I have found that students who cannot fathom the general rule \((\frac{a^m}{a^n} = a^{m-n}) \) often understand for the first time when they see concrete examples.
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Question 5 of 16
Observe the following equivalence:
42 43= 4 ×4 4 ×4 ×4It should be readily apparent that we may cancel out two quantities from both top and bottom of the fraction, so in the end we are left with this:
1 4Following the rule of \((\frac{a^m}{a^n} = a^{m-n})\), the reduction of \((\frac{4^2}{4^3})\) should be 4−1. Many students find this confusing, as the intuitive concept of exponents (how many times a number is to be multiplied by itself) fails here. How in the world do we multiply 4 by itself -1 times?!
Expand each of these expressions so that there are no exponents either:
\((\frac{3^2}{3^5})\) =
\((\frac{10^4}{10^6})\) =
\((\frac{8^3}{8^7})\) =
\((\frac{20^4}{20^5})\) =
- After expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (4−1), following the rule \((\frac{a^m}{a^n} = a^{m-n})\). From these examples, what easy-to-understand definition can you think of to describe negative exponents?
Also, expand the following expression so there are no exponents, then re-write the result in exponent form following the rule \((\frac{a^m}{a^n} = a^{m-n})\):
53 53What does this tell you about exponents of zero?
Reveal answerA negative exponent is simply the reciprocal (1/x) of its positive counterpart. A zero exponent is always equal to 1.
Notes:I have found that students who cannot fathom the meaning of negative or zero exponents often understand immediately when they construct their own definition based on the general rule \((\frac{a^m}{a^n} = a^{m-n})\).
- After expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (4−1), following the rule \((\frac{a^m}{a^n} = a^{m-n})\). From these examples, what easy-to-understand definition can you think of to describe negative exponents?
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Question 6 of 16
When evaluating (calculating) a mathematical expression, what order should you do the various expressions in? In other words, which comes first: multiplication, division, addition, subtraction, powers, roots, parentheses, etc.; and then what comes after that, and after that?
Reveal answerDo what is inside parentheses first (the furthest “inside” parentheses if there are multiple layers of parentheses), powers and roots, functions (trig, log, etc.), multiplication/division, and finally addition/subtraction.
Notes:Order of operations is extremely important, as it becomes critical to recognize proper order of evaluation when “stripping” an expression down to isolate a particular variable. In essence, the normal order of operations is reversed when “undoing” an expression, so students must recognize what the proper order of operations is.
Related Tools:
- Performance-Based Assessments for DC Circuit Competencies
- Conventional Transistor Overview and Special Transistors
Question 3, answer - the ‘+’ symbol is missing from ‘= a^(m+n)’.
Question 7, question and answer - second equation should have a multiplication symbol, not the ‘variable x’.
Question 8, question and answer - second equation should have a multiplication symbol, not the ‘variable x’.
Question 9, question - the ‘+’ symbol is missing between ‘4.5154 ‘+’ 14’.
Question 9, answer:
- the ‘+’ symbol is missing: 10 − 25 ×2 ‘+’ 5 = −35
- the ‘+’ symbol is missing: −8 ‘+’ 10^3 ×51 = 50992
- the ‘+’ symbol is missing: 12^4 ×(3 ‘+’ 11) = 290304
Question 9, question and answer: The square root should enclose the whole equation, and the equation should have a multiplication symbol, not the ‘variable x’.
Question 13, question and answer: The table columns need spacing and the ‘+’ symbol is missing from the headings ‘2x + 1’.
These are all correct in the PDF version.